Nonlinear stability of periodic traveling wave solutions of viscous conservation laws in dimensions one and two

TL;DR

This paper proves the nonlinear stability and describes the long-term behavior of periodic traveling wave solutions in viscous conservation laws specifically in one and two dimensions, extending previous higher-dimensional results.

Contribution

It introduces new analytical techniques for low-dimensional cases, including refined Green function estimates and a novel cancellation scheme in nonlinear analysis.

Findings

Established nonlinear stability in 1D and 2D cases.

Developed refined Green function estimates for these dimensions.

Introduced a new scheme for nonlinear cancellation detection.

Abstract

Extending results of Oh and Zumbrun in dimensions $d\ge 3$, we establish nonlinear stability and asymptotic behavior of spatially-periodic traveling-wave solutions of viscous systems of conservation laws in critical dimensions $d=1,2$, under a natural set of spectral stability assumptions introduced by Schneider in the setting of reaction diffusion equations. The key new steps in the analysis beyond that in dimensions $d\ge 3$ are a refined Green function estimate separating off translation as the slowest decaying linear mode and a novel scheme for detecting cancellation at the level of the nonlinear iteration in the Duhamel representation of a modulated periodic wave.